Statistical Procedure

P-Value, Significance Level, and Test Statistic Explained

A decision-and-workflow guide for p-values, significance levels, and test statistics, covering method selection, conditions, mathematics, calculator evidence, and contextual reporting.

Course status: Revised 2026-27 course
Updated: July 18, 2026
Practice: Easy, Tough and Toughest

Method at a Glance: P-Value Hypothesis Testing

A p-value is the probability, assuming H0 and the test model, of a statistic at least as extreme as observed in the alternative's direction; it is not P(H0 is true).

Reader tasktail area under H0, evidence strength, alpha decisions, and interpretation

Planned modules8

Mathematics1 expressions

Worked checks57

Boundary: A p-value is not the probability that H0 is true.

Procedure Workflow

Identify the data structure and parameter before selecting p value hypothesis testing; the name of a calculator menu is not method evidence.
State the hypotheses or estimation target for p value hypothesis testing using population notation and the order defined by the question.
Verify the design, independence, and approximation conditions that specifically justify p value hypothesis testing rather than reciting every condition learned in the course.
Compute the statistic, standard error, interval, or p-value for p value hypothesis testing with defined symbols, guard digits, and an independent arithmetic check.
Interpret p value hypothesis testing in the population and units named by the problem, then limit causation and generalization to what the collection design supports.

Procedure Formulas and Notation

P-value definition

p\text{-value}=P(\text{statistic at least as extreme}\mid H_0)

P-value definition in P-Value Hypothesis Testing: Define the event or random variable first; complements, conditioning, and trial assumptions determine which probability expression applies.

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Step 1

Test statistic

Decision

For Test statistic in p value hypothesis testing, A two-sided test for a seedling-growth comparison gives $z=1.40$ . Compute and interpret the p-value for test statistic.

Test statistic result in p value hypothesis testing: The two-sided p-value is approximately 0.1615.

p=2P(Z\ge|1.40|)=2[1-\Phi(1.40)]=0.1615.

Interpretation and validity

Test statistic interpretation for p value hypothesis testing: Make the alpha comparison, state reject or fail to reject H0, and conclude in terms of evidence for the population claim.

Condition evidence for p value hypothesis testing and Test statistic: The test statistic must be calculated under H0 and the two-sided direction must have been specified by the alternative.

Procedure error: The p-value is not the probability H0 is true; it is a conditional tail probability assuming H0.

Step 2

Null distribution

Decision

For Null distribution in p value hypothesis testing, A two-sided test for a package-delivery sample gives $z=1.65$ . Compute and interpret the p-value for null distribution.

Null distribution result in p value hypothesis testing: The two-sided p-value is approximately 0.0989.

p=2P(Z\ge|1.65|)=2[1-\Phi(1.65)]=0.0989.

Interpretation and validity

Null distribution interpretation for p value hypothesis testing: Make the alpha comparison, state reject or fail to reject H0, and conclude in terms of evidence for the population claim.

Condition evidence for p value hypothesis testing and Null distribution: The test statistic must be calculated under H0 and the two-sided direction must have been specified by the alternative.

Procedure error: The p-value is not the probability H0 is true; it is a conditional tail probability assuming H0.

Step 3

P-value

Decision

For P-value in p value hypothesis testing, A two-sided test for a classroom memory study gives $z=1.90$ . Compute and interpret the p-value for p-value.

P-value result in p value hypothesis testing: The two-sided p-value is approximately 0.0574.

p=2P(Z\ge|1.90|)=2[1-\Phi(1.90)]=0.0574.

Interpretation and validity

P-value interpretation for p value hypothesis testing: Make the alpha comparison, state reject or fail to reject H0, and conclude in terms of evidence for the population claim.

Condition evidence for p value hypothesis testing and P-value: The test statistic must be calculated under H0 and the two-sided direction must have been specified by the alternative.

Procedure error: The p-value is not the probability H0 is true; it is a conditional tail probability assuming H0.

Step 4

Significance level α

Decision

For Significance level α in p value hypothesis testing, A two-sided test for a water-filtration experiment gives $z=2.15$ . Compute and interpret the p-value for significance level α.

Significance level α result in p value hypothesis testing: The two-sided p-value is approximately 0.0316.

p=2P(Z\ge|2.15|)=2[1-\Phi(2.15)]=0.0316.

Interpretation and validity

Significance level α interpretation for p value hypothesis testing: Make the alpha comparison, state reject or fail to reject H0, and conclude in terms of evidence for the population claim.

Condition evidence for p value hypothesis testing and Significance level α: The test statistic must be calculated under H0 and the two-sided direction must have been specified by the alternative.

Procedure error: The p-value is not the probability H0 is true; it is a conditional tail probability assuming H0.

Step 5

Statistical significance

Decision

For Statistical significance in p value hypothesis testing, A two-sided test for a public-parks visitor survey gives $z=2.40$ . Compute and interpret the p-value for statistical significance.

Statistical significance result in p value hypothesis testing: The two-sided p-value is approximately 0.0164.

p=2P(Z\ge|2.40|)=2[1-\Phi(2.40)]=0.0164.

Interpretation and validity

Statistical significance interpretation for p value hypothesis testing: Make the alpha comparison, state reject or fail to reject H0, and conclude in terms of evidence for the population claim.

Condition evidence for p value hypothesis testing and Statistical significance: The test statistic must be calculated under H0 and the two-sided direction must have been specified by the alternative.

Procedure error: The p-value is not the probability H0 is true; it is a conditional tail probability assuming H0.

Step 6

Decision rules

Decision

For Decision rules in p value hypothesis testing, A two-sided test for a greenhouse germination experiment gives $z=2.65$ . Compute and interpret the p-value for decision rules.

Decision rules result in p value hypothesis testing: The two-sided p-value is approximately 0.0080.

p=2P(Z\ge|2.65|)=2[1-\Phi(2.65)]=0.0080.

Interpretation and validity

Decision rules interpretation for p value hypothesis testing: Make the alpha comparison, state reject or fail to reject H0, and conclude in terms of evidence for the population claim.

Condition evidence for p value hypothesis testing and Decision rules: The test statistic must be calculated under H0 and the two-sided direction must have been specified by the alternative.

Procedure error: The p-value is not the probability H0 is true; it is a conditional tail probability assuming H0.

Step 7

Strength of evidence

Decision

For Strength of evidence in p value hypothesis testing, A two-sided test for a quality-control inspection gives $z=1.40$ . Compute and interpret the p-value for strength of evidence.

Strength of evidence result in p value hypothesis testing: The two-sided p-value is approximately 0.1615.

p=2P(Z\ge|1.40|)=2[1-\Phi(1.40)]=0.1615.

Interpretation and validity

Strength of evidence interpretation for p value hypothesis testing: Make the alpha comparison, state reject or fail to reject H0, and conclude in terms of evidence for the population claim.

Condition evidence for p value hypothesis testing and Strength of evidence: The test statistic must be calculated under H0 and the two-sided direction must have been specified by the alternative.

Procedure error: The p-value is not the probability H0 is true; it is a conditional tail probability assuming H0.

Step 8

Interpretation errors

Decision

For Interpretation errors in p value hypothesis testing, A two-sided test for a commuter route study gives $z=1.65$ . Compute and interpret the p-value for interpretation errors.

Interpretation errors result in p value hypothesis testing: The two-sided p-value is approximately 0.0989.

p=2P(Z\ge|1.65|)=2[1-\Phi(1.65)]=0.0989.

Interpretation and validity

Interpretation errors interpretation for p value hypothesis testing: Make the alpha comparison, state reject or fail to reject H0, and conclude in terms of evidence for the population claim.

Condition evidence for p value hypothesis testing and Interpretation errors: The test statistic must be calculated under H0 and the two-sided direction must have been specified by the alternative.

Procedure error: The p-value is not the probability H0 is true; it is a conditional tail probability assuming H0.

Procedure Practice and Full Solutions

Every question in P-Value, Significance Level, and Test Statistic Explained is newly written from the revised framework and the logic visible in public College Board materials. Constructed numerical settings are identified as instructional scenarios and are never represented as measurements from a real population. No released or secure question wording is reproduced.

Easy Practice

Easy 1: Interpretation errors

Question P62-Easy-1. A two-sided test for a school library checkout study gives $z=1.40$ . Compute and interpret the p-value for interpretation errors.